Azula treats her like dirt all the time, same as she does with like, every other character.
This is true, but her treatment of Ty Lee in that particular episode always stuck out to me. It was just so pointlessly vindictive and mean-spirited. Watching her take shots at Ty Lee for simply having the nerve to succeed where she couldn't was devastating (I may be exaggerating slightly).
and it's Ty Lee's betrayal more than anything else that seems to send her really over the edge.
Eh, this seems like a stretch, though. She piggybacked pretty hard off of Mai's initial betrayal. It was clear just from Azula's expression and tone immediately prior to attacking Mai that she was already struggling to cope with what was happening. And maybe my memory's failing me, but I can't recall any scene after the fact that specifically emphasized Ty Lee's role in those events.
As for the rest, I can see your point, but I don't have to like it!